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प्रश्न
A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.
उत्तर
Let the radius of the sphere be 'r1'.
Let the radius of the hemisphere be 'r2'
TSA of sphere = `4pi r_1^2`
TSA of hemisphere = `3pi r_2^2`
TSA of sphere = TSA of hemi-sphere
`4pir_1^2 = 3pir_2^2`
`=> r_2^2 = 4/3r_1^2`
`=> r_2 = 2/sqrt3r_1`
Volume of sphere, `v_1 = 4/3pir_1^3`
Volume of hemisphere, `v_2 = 2/3pir_2^3`
`v_2 = 2/3pir_2^3`
`=> v_2 = 2/3pi((r_1 2)/(3sqrt3))^3`
`=> v_2 = 2/3pi(r_2^3 8)/(3sqrt3)`
Dividing v1 by v2
`v_1/v_2 = (4/3pir_1^3)/(2/3pi 8/(3sqrt3)r_1^3`
`=> v_1/v_2 = (4/3)/(2/3 8/(3sqrt3)`
`=> v_1/v_2 = 4/3 xx (9sqrt3)/16`
`=> v_1/v_2 = (3sqrt3)/4`
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